Enumeration of inequivalent irreducible Goppa codes

We consider irreducible Goppa codes over F_q of length qⁿ defined by polynomials of degree r, where q is a prime power and n,r are arbitrary positive integers. We obtain an upper bound on the number of such codes.     

Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 ⁿ generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n?1 primitive idempotents ofR ₂ ⁿ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n?1 idempotents are also obtained. The casen=2, 3 is dealt separately.     

On the prime power factorization of n!

For a fixed prime q, let eq(n) denote the order of q in the prime factorization of n!. For two fixed integers m?2 and r with 0?r?m−1, let A(x;m,q,r) denote the numbers of positive integers n?x for which . In this paper we shall prove a sharp asymptotic formula of A(x;m,q,r).     

Negacyclic duadic codes

This paper extends the concepts from cyclic duadic codes to negacyclic codes over F_q (q an odd prime power) of oddly even length. Generalizations of defining sets, multipliers, splittings, even-like and odd-like codes are given. Necessary and sufficient conditions are given for the existence of self-dual negacyclic codes over F_q and the existence of splittings of 2N, where N is odd. Other negacyclic codes can be extended by two coordinates in a way to create self-dual codes with familiar parameters.     

Poisson statistics via the Chinese Remainder Theorem

We consider the distribution of spacings between consecutive elements in subsets of Z/qZ, where q is highly composite and the subsets are defined via the Chinese Remainder Theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q has Poisson spacings. We also study the spacings of subsets of Z/q₁q₂Z that are created via the Chinese Remainder Theorem from subsets of Z/q₁Z and Z/q₂Z (for q₁,q₂ coprime), and give criteria for when the spacings modulo q₁q₂ are Poisson. Moreover, we also give some examples when the spacings modulo q₁q₂ are not Poisson, even though the spacings modulo q₁ and modulo q₂ are both Poisson.     

On the distribution of the orders of 2(mod u) for odd u

In this note we investigate the distribution of odd numbers u such that the order of 2(mod u) is not divisible by qⁿ, where q is an odd prime and Received: 1 July 2004     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

Some constructions of linearly optimal group codes

We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n,n-3,3]_q-codes over F_q for n=2q and n=3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed-Solomon codes as ideals of the group ring F_qG where G is an elementary abelian group of order q.     

On the composition length of finite primitive linear groups

LetG be a finite primitive linear group over a fieldK, whereK is a finite field or a number field. We bound the composition length ofG in terms of the dimension of the underlying vector space and of the degree ofK over its prime subfield. As a byproduct, we prove a result of number theory which bounds the number of prime factors (counting multiplicities) ofq ⁿ−1, whereq, n>1 are integers, improving a result of Turull and Zame [6].     

Latin hypercubes and MDS codes

Maximum distance separable (MDS) codes have special properties that give them excellent error correcting capabilities. Counting the number of q-ary MDS codes of length n and distance d, denoted by D_q(n,d)_MDS, is a very hard problem. This paper shows that for d=2, it amounts to counting the number of (n-1)-dimensional Latin hypercubes of order q. Thus, D_q(3,2)_MDS is the number of Latin squares of order q, which is known only for a few values of q. This paper proves constructively that D₃(n,2)_MDS=6·2^n-2.     

On subword decomposition and balanced polynomials

Let H(x) be a monic polynomial over a finite field F=GF(q). Denote by Na(n) the number of coefficients in Hn which are equal to an element a∈F, and by G the set of elements a∈F^× such that Na(n)>0 for some n. We study the relationship between the numbers (Na(n))_a∈G and the patterns in the base q representation of n. This enables us to prove that for “most” n's we have Na(n)≈Nb(n), a,b∈G. Considering the case H=x+1, we provide new results on Pascal's triangle modulo a prime. We also provide analogous results for the triangle of Stirling numbers of the first kind.     

Partitions of the set of natural numbers and their representation functions

For a given set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. In this paper we give a simple proof to two results by Sándor.     

Minimal quadratic residue cyclic codes of lengthp n(p odd prime)

The explicit expressions for the 2n + 1 primitive idempotents in $R_{p^ - } = F[x]/< x^{p^ - } - 1 > $ , whereF is the field of prime power orderq and the multiplicative order ofq modulop ⁿ is ?(p ⁿ)/2 (n ≥ 1 andp is an odd prime), are obtained. An algorithm for computing the generating polynomials of the minimal QR cyclic codes of lengthp ⁿ, generated by these primitive idempotents, is given and hence some bounds on the minimum distance of some QR codes of prime length overGF(q)(q = 2, 3, ...) are obtained.     

Über die Verteilung der primen quadratischen Reste und Nicht-Reste modp

LetP be an odd prime, denote byp _n (q _n ) then ^th prime P with (p _n /P)=1(=–1),d _n =q _n –p _n . We discuss the question whetherd _n changes sign infinitely often or not. Without using Turán's power sum method the following theorem is proved. Suppose that the L-function L(s, ), defined by the real primitive character modP, has no real root with 1/2<<1. Then the numbersd _n change sign infinitely often. The hypothesis is known to be true for allP with 2<P227 (J. B. Rosser. J. of Research of the Nat. Bureau of Standards 45, 505–514 (1950)).

     

On the prime power factorization of n!, II

Let p, q be primes and m be a positive integer. For a positive integer n, let ep(n) be the nonnegative integer with pep(n)|n and pep(n)+1?n. The following results are proved: (1) For any positive integer m, any prime p and any ε∈Zm, there are infinitely many positive integers n such that ; (2) For any positive integer m, there exists a constant D(m) such that if ε,δ∈Zm and p, q are two distinct primes with max{p,q}?D(m), then there exist infinitely many positive integers n such that , . Finally we pose four open problems.     

Computational results of duadic double circulant codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called duadic double circulant codes, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb{F}_{2}$ , $\mathbb{F}_{3}$ , $\mathbb{F}_{4}$ , $\mathbb{F}_{5}$ , and $\mathbb{F}_{7}$ . In particular, we find a new ternary self-dual [76,38,18] code and easily rediscover optimal binary self-dual codes with parameters [66,33,12], [68,34,12], [86,43,16], and [88,44,16] as well as a formally self-dual binary [82,41,14] code.     

Covering all triples on n marks by disjoint Steiner systems

Let q be a number all whose prime factors divide integers of the form 2^s ? 1, s odd. If n = q + 2, the (₃ⁿ) triples on n marks can be partitioned into q sets, each forming a Steiner triple system.     

Cyclotomy and its application to duadic codes

In this paper we treat cyclotomic binary duadic codes. The conjecture of Ding and Pless is that there are infinitely many cyclotomic duadic codes of prime lengths that are not quadratic residue codes. We shall prove this conjecture by using the special case of Tschebotareff's density theorem.     

Sur un problème de Rényi

Let (n) be the number of all prime divisors ofn and (n) the number of distinct prime divisors ofn. We definev _q (x)=|{nx(n)–(n)=q}|. In this paper, we give an asymptotic development ofv _q (x); this improves on previous results.

     

Partitions of natural numbers with the same representation functions

For a set A of nonnegative integers the representation functions R₂(A,n), R₃(A,n) are defined as the number of solutions of the equation n=a+a^′,a,a^′∈A with a<a^′, a?a^′, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A₀ be the set of all nonnegative integers a with even D(a) and A₁ be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R₂(A,n)=R₂(N?A,n) for all n?2N−1, then R₂(A,n)=R₂(N?A,n)?1 for all n?12N²−10N−2 except for A=A₀ or A=A₁; (b) if R₃(A,n)=R₃(N?A,n) for all n?2N−1, then R₃(A,n)=R₃(N?A,n)?1 for all n?12N²+2N. Several problems are posed in this paper.